Reconstructing convex polyominoes from horizontal and vertical projections

Barcucci, Elena; Lungo, Alberto Del; Nivat, Maurice; Pinzani, Renzo

doi:10.1016/0304-3975(94)00293-2

Cited by 165 publications

(164 citation statements)

References 15 publications

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“…Woeginger [10] proved that the reconstruction problem in the class of polyominoes is an NP-complete problem. Barcucci, Del Lungo, Nivat, Pinzani [1] showed that the reconstruction problem is also NP-complete in the class of hconvex polyominoes and in the class v-convex polyominoes.…”

Section: Previous Resultsmentioning

confidence: 99%

The Reconstruction of Some 3D Convex Polyominoes from Orthogonal Projections

Gębala

2002

SOFSEM 2002: Theory and Practice of Informatics

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Section: Previous Resultsmentioning

confidence: 99%

The Reconstruction of Some 3D Convex Polyominoes from Orthogonal Projections

Gębala

2002

SOFSEM 2002: Theory and Practice of Informatics

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“…For example, it would be interesting to study how a reconstruction algorithm could work if the image must represent a convex polyomino as in [4], or more generally if we have of some skeletal information as in [15]. Introducing the right assumptions, this line of research could indeed obtain results that could be applied in tomographic applications dealing with reconstruction artifacts caused by blocking components.…”

Section: Discussionmentioning

confidence: 99%

“…To tackle these problems and to include some other information that may model effectively properties of the image to be rebuilt, often discrete tomography problems include some prior knowledge that gives birth to many variations of the reconstruction problem. Some examples that have been studied in literature include connectivity and convexity [4,2], cell coloring [10,3] or skeletal properties [14,15]. Often, with appropriate assumptions, the arising problems result to be connected to other fields of study as timetabling [18], image compression [1], network flow [5], graph theory [7] and combinatorics [13].…”

Section: Introductionmentioning

confidence: 99%

Discrete Tomography Reconstruction Algorithms for Images with a Blocking Component

Bilotta

Brocchi

2014

Lecture Notes in Computer Science

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Abstract. We study a problem involving the reconstruction of an image from its horizontal and vertical projections in the case where some parts of these projections are unavailable. The desired goal is to model applications where part of the x-rays used for the analysis of an object are blocked by particularly dense components that do not allow the rays to pass through the material. This is a common issue in many tomographic scans, and while there are several heuristics to handle quite efficiently the problem in applications, the underlying theory has not been extensively developed. In this paper, we study the properties of consistency and uniqueness of this problem, and we propose an efficient reconstruction algorithm. We also show how this task can be reduced to a network flow problem, similarly to the standard reconstruction algorithm, allowing the determination of a solution even in the case where some pixels of the output image must have some prescribed values.

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“…The cells set corresponding to a 4-connected discrete set is called polyomino. They are well-known combinatorial objects [9,10] and are related to many different problems, such as: tiling [2,5], enumeration [3,13] and discrete tomography [1,8]. In [11], the authors studied the problem of determining the medians of a polyomino, by considering a polyomino as a graph and by using the classical metric on the graph.…”

Section: -Pathmentioning

confidence: 99%

Medians of Discrete Sets according to a Linear Distance

Daurat¹,

Lungo²,

Nivat

2000

Discrete Comput Geom

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In this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions.

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Reconstructing convex polyominoes from horizontal and vertical projections

Cited by 165 publications

References 15 publications

The Reconstruction of Some 3D Convex Polyominoes from Orthogonal Projections

The Reconstruction of Some 3D Convex Polyominoes from Orthogonal Projections

Discrete Tomography Reconstruction Algorithms for Images with a Blocking Component

Medians of Discrete Sets according to a Linear Distance

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